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12
ANALYTICAL MECHANICS
ILLUSTRATIVE EXAMPLE.
A man walks 3 miles N. 30 E., then one mile E., then 3 miles S. 45 R,, then 4 miles S., then one mile N. 30 W. Find his final position.
Representing the displacements by vectors we obtain the graphical solution given in Fig. 13, where r represents the resultant displacement.
r\
FIG. 13.
In order to find r analytically we first determine its components.   Thuus
rx = [3 cos (60) + cos (0) + 2 cos (-45) + 4 COM (-90)
+ cos (120)] miles = (2 + V5) miles  3.41 miles. ry = [3 sin (60) + sin (0) + 2 sin (-45) + 4 sin (-90)
+ sin (120)] miles = (2V3~V2-4)miles = 1,95 miles.
= 3.93 miles. The direction of r is given by the following relation.
,    Q    rv.. -1.95 tan^^ = T4T'
/. e = -37.i.
Therefore the final position of the man is about 3.93 miles S. 52,!) R from his starting point.